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Symmetry
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Advances in Mathematical Models and Partial Differential Equations
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Advances in Mathematical Models and Partial Differential Equations

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 30 April 2024 | Viewed by 1429

Special Issue Editors

Prof. Jianwei Dong

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Guest Editor
School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450046, China
Interests: linear algebra
Prof. Dr. Ningan Lai

E-Mail Website
Guest Editor
College of mathematical and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
Interests: partial differential equations in fluid mechanics; hyperbolic partial differential equation
Dr. Jianli Liu

E-Mail Website
Guest Editor
Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: apply partial differential equations
Dr. Manwai Yuen

E-Mail Website
Guest Editor
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong
Interests: partial differential equations; symmetry reduction; blowup; Euler-Poisson equations; Euler equations with or without Coriolis Force; Camassa-Holm equations; Navier-Stokes equations; Magnetohydrodynamics (MHD); analytical and exact solutions; mathematical methods in fluids; classical cosmology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the study of partial differential equations (PDE), “blow-up” or “singularity” means the breakdown of a system within a finite time. The singularity formation in nonlinear physical systems has been attracting the attention of many physics and mathematics researchers because of its physical significance and mathematical challenge. In this regard, the lifespan of a PDE system is the maximal time before which the solutions exist and are sufficiently smooth.

In the study of PDE, it is expected that symmetry assumptions or reductions can facilitate the study of the lifespan of the non-linear partial differential systems. In order words, symmetry is especially useful to analyze simpler cases of some complex systems.

In this Special Issue, we expect that theoretical or numerical study of the lifespan of non-linear PDE, can be developed. To contribute to this Special Issue, we expect that the theoretical analysis can establish a sufficient condition on initial data that guarantees that the lifespan of the systems is finite. For the numerical study of the lifespan problem, the maximal existence time must be estimated with significant improvement.

Prof. Jianwei Dong
Prof. Dr. Ningan Lai
Dr. Jianli Liu
Dr. Manwai Yuen
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • partial differential equations
  • lifespan
  • global existence
  • blowup
  • symmetry reduction
  • exact solutions
  • smoothness
  • regularity

Published Papers (2 papers)

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Research

19 pages, 328 KiB  
Article
Formation of Singularity for Isentropic Irrotational Compressible Euler Equations
by Jianli Liu, Ziyi Qin and Manwai Yuen
Symmetry 2024, 16(4), 454; https://doi.org/10.3390/sym16040454 - 08 Apr 2024
Viewed by 376
Abstract
The domain of science and engineering relies heavily on an in-depth comprehension of fluid dynamics, given the prevalence of fluids such as water, air, and interstellar gas in the universe. Euler equations form the basis for the study of fluid motion. This paper [...] Read more.
The domain of science and engineering relies heavily on an in-depth comprehension of fluid dynamics, given the prevalence of fluids such as water, air, and interstellar gas in the universe. Euler equations form the basis for the study of fluid motion. This paper is concerned with the Cauchy problem of isentropic compressible Euler equations away from the vacuum. We use the integration method with the general test function f=f(r), proving that there exist the corresponding blowup results of C1 irrotational solutions for Euler equations and Euler equations with time-dependent damping in Rn (n≥2), provided the density-independent initial functional is sufficiently large. We also provide two simple and explicit test functions f(r)=r and f(r)=1+r, to demonstrate the blowup phenomenon in the one-dimensional case. In particular, our results are applicable to the non-radial system. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations)
17 pages, 587 KiB  
Article
Symmetric-Type Multi-Step Difference Methods for Solving Nonlinear Equations
by Ioannis K. Argyros, Stepan Shakhno, Samundra Regmi, Halyna Yarmola and Michael I. Argyros
Symmetry 2024, 16(3), 330; https://doi.org/10.3390/sym16030330 - 08 Mar 2024
Viewed by 585
Abstract
Symmetric-type methods (STM) without derivatives have been used extensively to solve nonlinear equations in various spaces. In particular, multi-step STMs of a higher order of convergence are very useful. By freezing the divided differences in the methods and using a weight operator a [...] Read more.
Symmetric-type methods (STM) without derivatives have been used extensively to solve nonlinear equations in various spaces. In particular, multi-step STMs of a higher order of convergence are very useful. By freezing the divided differences in the methods and using a weight operator a method is generated using m steps (m a natural number) of convergence order 2 m. This method avoids a large increase in the number of operator evaluations. However, there are several problems with the conditions used to show the convergence: the existence of high order derivatives is assumed, which are not in the method; there are no a priori results for the error distances or information on the uniqueness of the solutions. Therefore, the earlier studies cannot guarantee the convergence of the method to solve nondifferentiable equations. However, the method may converge to the solution. Thus, the convergence conditions can be weakened. These problems arise since the convergence order is determined using the Taylor series which requires the existence of high-order derivatives which are not present in the method, and they may not even exist. These concerns are our motivation for authoring this article. Moreover, the novelty of this article is that all the aforementioned problems are addressed positively, and by using conditions only related to the divided differences in the method. Furthermore, a more challenging and important semi-local analysis of convergence is presented utilizing majorizing sequences in combination with the concept of the generalized continuity of the divided difference involved. The convergence is also extended from the Euclidean to the Banach space. We have chosen to demonstrate our technique in the present method. But it can be used in other studies using the Taylor series to show the convergence of the method. The applicability of other single- or multi-step methods using the inverses of linear operators with or without derivatives can also be extended with the same methodology along the same lines. Several examples are provided to test the theoretical results and validate the performance of the method. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations)
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